\(\int \frac {(a+b \log (c x^n))^3}{x^2} \, dx\) [62]

Optimal result

Integrand size = 16, antiderivative size = 69 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {6 b^3 n^3}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \]

[Out]

-6*b^3*n^3/x-6*b^2*n^2*(a+b*ln(c*x^n))/x-3*b*n*(a+b*ln(c*x^n))^2/x-(a+b*ln(c*x^n))^3/x

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {6 b^3 n^3}{x} \]

[In]

Int[(a + b*Log[c*x^n])^3/x^2,x]

[Out]

(-6*b^3*n^3)/x - (6*b^2*n^2*(a + b*Log[c*x^n]))/x - (3*b*n*(a + b*Log[c*x^n])^2)/x - (a + b*Log[c*x^n])^3/x

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x}+(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx \\ & = -\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x}+\left (6 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx \\ & = -\frac {6 b^3 n^3}{x}-\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^3+3 b n \left (\left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (a+b n+b \log \left (c x^n\right )\right )\right )}{x} \]

[In]

Integrate[(a + b*Log[c*x^n])^3/x^2,x]

[Out]

-(((a + b*Log[c*x^n])^3 + 3*b*n*((a + b*Log[c*x^n])^2 + 2*b*n*(a + b*n + b*Log[c*x^n])))/x)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.64

method	result	size
parallelrisch	\(-\frac {b^{3} \ln \left (c \,x^{n}\right )^{3}+3 \ln \left (c \,x^{n}\right )^{2} b^{3} n +6 \ln \left (c \,x^{n}\right ) b^{3} n^{2}+6 b^{3} n^{3}+3 a \,b^{2} \ln \left (c \,x^{n}\right )^{2}+6 \ln \left (c \,x^{n}\right ) a \,b^{2} n +6 a \,b^{2} n^{2}+3 a^{2} b \ln \left (c \,x^{n}\right )+3 a^{2} b n +a^{3}}{x}\)	\(113\)
risch	\(\text {Expression too large to display}\)	\(2674\)

[In]

int((a+b*ln(c*x^n))^3/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/x*(b^3*ln(c*x^n)^3+3*ln(c*x^n)^2*b^3*n+6*ln(c*x^n)*b^3*n^2+6*b^3*n^3+3*a*b^2*ln(c*x^n)^2+6*ln(c*x^n)*a*b^2*
n+6*a*b^2*n^2+3*a^2*b*ln(c*x^n)+3*a^2*b*n+a^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (69) = 138\).

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.61 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {b^{3} n^{3} \log \left (x\right )^{3} + 6 \, b^{3} n^{3} + b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 3 \, a^{2} b n + a^{3} + 3 \, {\left (b^{3} n + a b^{2}\right )} \log \left (c\right )^{2} + 3 \, {\left (b^{3} n^{3} + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )^{2} + 3 \, {\left (2 \, b^{3} n^{2} + 2 \, a b^{2} n + a^{2} b\right )} \log \left (c\right ) + 3 \, {\left (2 \, b^{3} n^{3} + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + a^{2} b n + 2 \, {\left (b^{3} n^{2} + a b^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{x} \]

[In]

integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="fricas")

[Out]

-(b^3*n^3*log(x)^3 + 6*b^3*n^3 + b^3*log(c)^3 + 6*a*b^2*n^2 + 3*a^2*b*n + a^3 + 3*(b^3*n + a*b^2)*log(c)^2 + 3
*(b^3*n^3 + b^3*n^2*log(c) + a*b^2*n^2)*log(x)^2 + 3*(2*b^3*n^2 + 2*a*b^2*n + a^2*b)*log(c) + 3*(2*b^3*n^3 + b
^3*n*log(c)^2 + 2*a*b^2*n^2 + a^2*b*n + 2*(b^3*n^2 + a*b^2*n)*log(c))*log(x))/x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (63) = 126\).

Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=- \frac {a^{3}}{x} - \frac {3 a^{2} b n}{x} - \frac {3 a^{2} b \log {\left (c x^{n} \right )}}{x} - \frac {6 a b^{2} n^{2}}{x} - \frac {6 a b^{2} n \log {\left (c x^{n} \right )}}{x} - \frac {3 a b^{2} \log {\left (c x^{n} \right )}^{2}}{x} - \frac {6 b^{3} n^{3}}{x} - \frac {6 b^{3} n^{2} \log {\left (c x^{n} \right )}}{x} - \frac {3 b^{3} n \log {\left (c x^{n} \right )}^{2}}{x} - \frac {b^{3} \log {\left (c x^{n} \right )}^{3}}{x} \]

[In]

integrate((a+b*ln(c*x**n))**3/x**2,x)

[Out]

-a**3/x - 3*a**2*b*n/x - 3*a**2*b*log(c*x**n)/x - 6*a*b**2*n**2/x - 6*a*b**2*n*log(c*x**n)/x - 3*a*b**2*log(c*
x**n)**2/x - 6*b**3*n**3/x - 6*b**3*n**2*log(c*x**n)/x - 3*b**3*n*log(c*x**n)**2/x - b**3*log(c*x**n)**3/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {b^{3} \log \left (c x^{n}\right )^{3}}{x} - 3 \, {\left (2 \, n {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} + \frac {n \log \left (c x^{n}\right )^{2}}{x}\right )} b^{3} - 6 \, a b^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b n}{x} - \frac {3 \, a^{2} b \log \left (c x^{n}\right )}{x} - \frac {a^{3}}{x} \]

[In]

integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="maxima")

[Out]

-b^3*log(c*x^n)^3/x - 3*(2*n*(n^2/x + n*log(c*x^n)/x) + n*log(c*x^n)^2/x)*b^3 - 6*a*b^2*(n^2/x + n*log(c*x^n)/
x) - 3*a*b^2*log(c*x^n)^2/x - 3*a^2*b*n/x - 3*a^2*b*log(c*x^n)/x - a^3/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (69) = 138\).

Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.86 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {b^{3} n^{3} \log \left (x\right )^{3}}{x} - \frac {3 \, {\left (b^{3} n^{3} + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )^{2}}{x} - \frac {3 \, {\left (2 \, b^{3} n^{3} + 2 \, b^{3} n^{2} \log \left (c\right ) + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n\right )} \log \left (x\right )}{x} - \frac {6 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \left (c\right ) + 3 \, b^{3} n \log \left (c\right )^{2} + b^{3} \log \left (c\right )^{3} + 6 \, a b^{2} n^{2} + 6 \, a b^{2} n \log \left (c\right ) + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b n + 3 \, a^{2} b \log \left (c\right ) + a^{3}}{x} \]

[In]

integrate((a+b*log(c*x^n))^3/x^2,x, algorithm="giac")

[Out]

-b^3*n^3*log(x)^3/x - 3*(b^3*n^3 + b^3*n^2*log(c) + a*b^2*n^2)*log(x)^2/x - 3*(2*b^3*n^3 + 2*b^3*n^2*log(c) +
b^3*n*log(c)^2 + 2*a*b^2*n^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)/x - (6*b^3*n^3 + 6*b^3*n^2*log(c) + 3*b^3*n*
log(c)^2 + b^3*log(c)^3 + 6*a*b^2*n^2 + 6*a*b^2*n*log(c) + 3*a*b^2*log(c)^2 + 3*a^2*b*n + 3*a^2*b*log(c) + a^3
)/x

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {a^3+3\,a^2\,b\,n+6\,a\,b^2\,n^2+6\,b^3\,n^3}{x}-\frac {\ln \left (c\,x^n\right )\,\left (3\,a^2\,b+6\,a\,b^2\,n+6\,b^3\,n^2\right )}{x}-\frac {b^3\,{\ln \left (c\,x^n\right )}^3}{x}-\frac {3\,b^2\,{\ln \left (c\,x^n\right )}^2\,\left (a+b\,n\right )}{x} \]

[In]

int((a + b*log(c*x^n))^3/x^2,x)

[Out]

- (a^3 + 6*b^3*n^3 + 6*a*b^2*n^2 + 3*a^2*b*n)/x - (log(c*x^n)*(3*a^2*b + 6*b^3*n^2 + 6*a*b^2*n))/x - (b^3*log(
c*x^n)^3)/x - (3*b^2*log(c*x^n)^2*(a + b*n))/x